In Team Semantics, a dependency notion is strongly first order if every sentence of the logic obtained by adding the corresponding atoms to First Order Logic is equivalent to some first order sentence. In this work it is shown that all nontrivial dependency atoms that are strongly first order, downwards closed, and relativizable (in the sense that the relativizations of the corresponding atoms with respect to some unary predicate are expressible in terms of them) are definable in terms of constancy atoms.Additionally, it is shown that any strongly first order dependency is safe for any family of downwards closed dependencies, in the sense that every sentence of the logic obtained by adding to First Order Logic both the strongly first order dependency and the downwards closed dependencies is equivalent to some sentence of the logic obtained by adding only the downwards closed dependencies.
Preliminaries
Team Semantics and DependenciesAs mentioned in the Introduction, Team Semantics generalizes Tarskian semantics for First Order Logic by letting formulas be satisfied or not satisfied by sets of assignments, which are called teams for historical reasons: Definition 2.1 (Team). Let M be a first order model (over any signature Σ) with domain M and let V be a finite set of variables. Then a team X over M with domain Dom(X) = V is a set of variable assignments s : V → M over M.There exists an obvious correspondence between teams and relations:Definition 2.2 (Teams to Relations). Let X be a team over some model M, and let t = t 1 . . . t n be a finite tuple of terms in the signature of M with variables in Dom(X). Then we write X(t) for the n-ary relationwhere each t M i (s) is the interpretation of t i in M for the assignment s.Teams can be restricted to a subset of the variables in their domain in the obvious way: Definition 2.3 (Team Restriction). Let X be a team over some model M and let V ⊆ Dom(X) be a subset of the variables in its domain. Then we write X |V for the restriction of X to the variables in V , that is forwhere, for all assignments s ∈ X, s |V is the unique assignment with domain V such that s |V (v) = s(v) for all variables v ∈ V .